cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A123610 Triangle read by rows, where T(n,k) = (1/n)*Sum_{d|(n,k)} phi(d) * binomial(n/d,k/d)^2 for n >= k > 0, with T(n,0) = 1 for n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 20, 20, 5, 1, 1, 6, 39, 68, 39, 6, 1, 1, 7, 63, 175, 175, 63, 7, 1, 1, 8, 100, 392, 618, 392, 100, 8, 1, 1, 9, 144, 786, 1764, 1764, 786, 144, 9, 1, 1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1, 1, 11, 275, 2475, 9900
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2006

Keywords

Comments

A variant of the triangle A047996 of circular binomial coefficients.

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  3,   3,    1;
  1,  4,  10,    4,    1;
  1,  5,  20,   20,    5,    1;
  1,  6,  39,   68,   39,    6,    1;
  1,  7,  63,  175,  175,   63,    7,    1;
  1,  8, 100,  392,  618,  392,  100,    8,   1;
  1,  9, 144,  786, 1764, 1764,  786,  144,   9,  1;
  1, 10, 205, 1440, 4420, 6352, 4420, 1440, 205, 10, 1;
  ...
Example of column g.f.s are:
column 1: 1/(1 - x)^2;
column 2: Ser([1, 1, 3, 1]) / ((1 - x)^2*(1 - x^2)^2) = g.f. of A005997;
column 3: Ser([1, 2, 11, 26, 30, 26, 17, 6, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 -x^3)^2);
column 4: Ser([1, 3, 28, 94, 240, 440, 679, 839, 887, 757, 550, 314, 148, 48, 11, 1]) / ((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2*(1 - x^4)^2);
where Ser() denotes a polynomial in x with the given coefficients, as in Ser([1, 1, 3, 1]) = (1 + x + 3*x^2 + x^3).

Links

Crossrefs

Cf. Columns: A005997, A123613, A123614, A123615, A123616.
Cf. A123611 (row sums), A123612 (antidiagonal sums), A123617 (central terms).
Cf. A123618, A123619, A047996 (variant), A128545.

Programs

  • Mathematica
    T[, 0] = 1; T[n, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]* Binomial[n/#, k/#]^2, 0]&]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2015, adapted from PARI *)
  • PARI
    {T(n,k)=if(k==0,1,(1/n)*sumdiv(n,d,if(gcd(k,d)==d, eulerphi(d)*binomial(n/d,k/d)^2,0)))}

Formula

T(2*n+1, n) = (2*n + 1)*A000108(n)^2 = (2*n + 1)*((2*n)!/(n!(n+1)!))^2 = A000891(n) for n >= 0.
Row sums are 2*A047996(2*n,n) = 2*A003239(n) for n > 0.
Row sums equal the row sums of triangle A128545.
For n >= 1, the g.f. of column n has the form: P_n(x)/(Product_{m=1..n} (1 - x^m)^2), where P_n(x) is a polynomial with n^2 coefficients such that the sum of the coefficients is P_n(1) = (2*n - 1)!.
From Petros Hadjicostas, Oct 24 2017: (Start)
Proofs of the following formulae can be found in the links.
G.f.: Sum_{n>=1, k>=0} T(n,k)*x^n*y^k = -Sum_{s>=1} (phi(s)/s)*log(f(x^s,y^s)), where phi(s) is Euler's totient function at s, f(x,y) = (sqrt(g(x,y)) + 1 -(1 + y)*x)/2, and g(x,y) = 1 - 2*(1 + y)*x + (1 - y)^2*x^2. (Term T(0,0) is not used in this g.f.)
Row g.f.: Sum_{k>=0} T(n,k)*y^k = (1/n)*Sum_{d|n} phi(d)*R(n/d, y^d), where R(m, y) = [z^m] (1 + (1 + y)*z + y*z^2)^m. (End)

A123613 Column 3 of triangle A123610.

Original entry on oeis.org

1, 4, 20, 68, 175, 392, 786, 1440, 2475, 4036, 6292, 9464, 13805, 19600, 27200, 36996, 49419, 64980, 84238, 107800, 136367, 170696, 211600, 260000, 316881, 383292, 460404, 549460, 651775, 768800, 902066, 1053184, 1223915, 1416108, 1631700, 1872792, 2141581
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2006

Keywords

Links

Crossrefs

Cf. A123610 (triangle); columns: A005997, A123614, A123615, A123616.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 2*x + 11*x^2 + 26*x^3 + 30*x^4 + 26*x^5 + 17*x^6 + 6*x^7 + x^8)/((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2), {x, 0, 50}], x] (* G. C. Greubel, Oct 16 2017 *)
LinearRecurrence[{4,-6,6,-9,12,-9,6,-6,4,-1},{1,4,20,68,175,392,786,1440,2475,4036},40] (* Harvey P. Dale, Apr 22 2019 *)
  • PARI
    {a(n)=polcoeff(truncate(Ser([1,2,11,26,30,26,17,6,1]))/((1-x)^2*(1-x^2)^2*(1-x^3)^2 +x*O(x^n)),n)}

Formula

G.f.: P_3(x) / ((1-x)^2*(1-x^2)^2*(1-x^3)^2), with P_3(1) = 5!, where P_3(x) = (1+2*x+11*x^2+26*x^3+30*x^4+26*x^5+17*x^6+6*x^7+x^8).

A123615 Column 5 of triangle A123610.

Original entry on oeis.org

1, 6, 63, 392, 1764, 6352, 19404, 52272, 127413, 286286, 601203, 1192464, 2252432, 4078368, 7116336, 12018704, 19718181, 31521798, 49228487, 75274584, 112911880, 166423400, 241382700, 344962800, 486301725, 676932006, 931282191, 1267259168
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2006

Keywords

Links

Crossrefs

Cf. A123610 (triangle); columns: A005997, A123613, A123614, A123616.

Programs

  • Mathematica
    CoefficientList[Series[(1 + 4*x + 50*x^2 + 262*x^3 + 930*x^4 + 2566*x^5 + 5795*x^6 + 11156*x^7 + 18699*x^8 + 27712*x^9 + 36699*x^10 + 43696*x^11 + 46988*x^12 + 45696*x^13 + 40167*x^14 + 31828*x^15 + 22603*x^16 + 14268*x^17 + 7899*x^18 + 3762*x^19 + 1498*x^20 + 474*x^21 + 110*x^22 + 16*x^23 + x^24)/((1 - x)^2*(1 - x^2)^2*(1 - x^3)^2*(1 - x^4)^2*(1 - x^5)^2), {x, 0, 50}], x] (* G. C. Greubel, Oct 16 2017 *)
  • PARI
    {a(n)=polcoeff(truncate(Ser([1,4,50,262,930,2566,5795,11156,18699,27712, 36699,43696,46988,45696,40167,31828,22603,14268,7899,3762,1498,474,110,16,1])) /((1-x)^2*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)^2 +x*O(x^n)),n)}

Formula

G.f.: P_5(x) / ((1-x)^2*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)^2), with P_5(1) = 9!, where P_5(x) = (1+4*x+50*x^2+262*x^3+930*x^4+2566*x^5+5795*x^6+11156*x^7+ 18699*x^8+27712*x^9+36699*x^10+43696*x^11+46988*x^12+45696*x^13+ 40167*x^14+31828*x^15+22603*x^16+14268*x^17+7899*x^18+3762*x^19+ 1498*x^20+474*x^21+110*x^22+16*x^23+x^24).

A123616 Column 6 of triangle A123610.

Original entry on oeis.org

1, 7, 100, 786, 4420, 19404, 71188, 226512, 644231, 1670015, 4008200, 9009728, 19146090, 38744496, 75117600, 140218218, 253051227, 443056383, 754838884, 1254576400, 2038689796, 3245256396, 5069041432, 7780827600, 11752298725
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2006

Keywords

Links

Crossrefs

Cf. A123610 (triangle); columns: A005997, A123613, A123614, A123615.

Programs

  • PARI
    {a(n)=polcoeff(truncate(Ser([1,5,85,581,2763,9987,29644,74546,164629, 324255,579250,946960,1429875,2003713,2620218,3205496,3679773,3967701, 4024087,3837087,3440204,2894878,2283089,1681653,1153208,731684,427027, 226843,108486,45806,16737,5073,1221,211,23,1])) / ((1-x)^2*(1-x^2)^2*(1-x^3)^2*(1-x^4)^2*(1-x^5)^2*(1-x^6)^2 +x*O(x^n)),n)}

Formula

G.f.: P_6(x) / ((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6))^2, with P_6(1) = 11!, where P_6(x) = (1+5*x+85*x^2+581*x^3+2763*x^4+9987*x^5+29644*x^6+74546*x^7+ 164629*x^8+324255*x^9+579250*x^10+946960*x^11+1429875*x^12+2003713*x^13+ 2620218*x^14+3205496*x^15+3679773*x^16+3967701*x^17+4024087*x^18+ 3837087*x^19+3440204*x^20+2894878*x^21+2283089*x^22+1681653*x^23+ 1153208*x^24+731684*x^25+427027*x^26+226843*x^27+108486*x^28+45806*x^29+ 16737*x^30+5073*x^31+1221*x^32+211*x^33+23*x^34+x^35).
Showing 1-4 of 4 results.

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